Definition 3.1.6 (Subbase of Topology).label Let $X$ be a set then $\cali{S}\suf\cali{P}(X)$ is a subbase if it is a covering of $X$. The smallest topology on $X$ containing $\cali{S}$ is given by
\begin{align*}\tau(\cali{S})\define\curl{\cups{}{i\in I }U_i:U_i\in\cali{B}(\cali{S})}=\caps{}{\text{topology }\tau\supf \cali{S}}\tau\end{align*}
where $\tau(\cali{S})$ is called the topology generated by $\cali{S}$ and
\begin{align*}\cali{B}(\cali{S})=\curl{\caps{}{j\in J }U_j:\curl{U_j}_{j\in J}\suf\cali{S},\abs{J}<\infty}\end{align*}
is a base for $\tau(\cali{S})$.
Proof. By construction $\cali{B}(\cali{S})$ is a base hence $\tau(\cali{S})$ is a topology by Definition 3.1.5 whose base is $\cali{B}(\cali{S})$. The inclusion $\tau(\cali{S})\suf \caps{}{\tau\supf \cali{S}}\tau$ follows from (O2) and (O3). On the other hand $\tau(\cali{S})$ is a topology that contains $\cali{S}$ so we are done.$\square$
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